Necessary and Sufficient No-Arbitrage Conditions for the SSVI/S3 Volatility Curve

Abstract

The "SSVI" (aka "S3") implied volatility curve is the simplest curve that has three parameters to describe the at-the-money behavior of implied volatilities for a given term, while also having a sensible functional form in the call and put wings. We describe the necessary and sufficient conditions on its three parameters to avoid butterfly arbitrage. By considering dimensionless parameters in normalized strike space, a simple picture of the no-arbitrage region emerges. For fixed (normalized) volatility it is compact, simply connected, and surprisingly large for realistic volatilities. For very large (normalized) volatility the asymptotic wing constraints of Lee are not just necessary but also sufficient for the absence of arbitrage everywhere.

Keywords: options, implied volatility, volatility curve parametrization, no-arbitrage, SSVI/S3

JEL Classification: G12, G13, C63